Legendre Deep Neural Network (LDNN) and its application for approximation of nonlinear Volterra Fredholm Hammerstein integral equations
This work addresses a specific problem in numerical analysis for researchers in applied mathematics and engineering, but it appears incremental as it combines existing methods like Gaussian quadrature with a new activation function.
The paper tackles solving nonlinear Volterra Fredholm Hammerstein integral equations by proposing the Legendre Deep Neural Network (LDNN), which uses Legendre orthogonal polynomials as activation functions, and demonstrates its performance and accuracy through several examples.
Various phenomena in biology, physics, and engineering are modeled by differential equations. These differential equations including partial differential equations and ordinary differential equations can be converted and represented as integral equations. In particular, Volterra Fredholm Hammerstein integral equations are the main type of these integral equations and researchers are interested in investigating and solving these equations. In this paper, we propose Legendre Deep Neural Network (LDNN) for solving nonlinear Volterra Fredholm Hammerstein integral equations (VFHIEs). LDNN utilizes Legendre orthogonal polynomials as activation functions of the Deep structure. We present how LDNN can be used to solve nonlinear VFHIEs. We show using the Gaussian quadrature collocation method in combination with LDNN results in a novel numerical solution for nonlinear VFHIEs. Several examples are given to verify the performance and accuracy of LDNN.